Accurate liver segmentation is important for a number of medical applications such as volumetric measurement prior to liver surgery or transplant, and for the creation of surfaces to be used in image-guided surgical systems. In the latter, surfaces created from pre-operative tomographic image volumes are registered to points acquired intra-operatively. Liver segmentation from both magnetic resonance (hereinafter “MR”) and computed tomography (hereinafter “CT”) images represents a significant segmentation challenge because the liver is surrounded by structures and tissues that have intensity values similar to its own and its shape is extremely variable. Despite these difficulties, several approaches have been proposed. For instance, it is understood that Bae et al [1] developed a method to sequentially segment the liver from abdomen cross-sections starting with a reference image. It is understood that the authors rely on classical image processing steps such as thresholding and connected component labeling followed by a post-processing stage in which a priori information on the liver morphology is used. Moreoever, it is understood that Gao et al [2] take a similar approach and use parametrically deformable contours. Other approaches based on sequences of image processing algorithms can be found in Chemouny et al [3], Inaoka et al [4], and Fishman et al [5]. In general, it is understood that these methods require a number of heuristics and parameter adjustments, which reduces their robustness. Montagnat and Delingette [6] have investigated the possibility of segmenting the liver from abdominal CT images by deforming a liver model, where a hybrid method that integrates global deformation and local deformation is employed. It is understood that the deformable model is thus shape constrained, and has been successfully applied in a limited number of livers. Recently, it is understood that Soler et al [7] have used this technique for automatic liver segmentation from spiral CT scans. In the approach, the skin, lungs, bones, kidneys, spleen, and liver are first extracted from the images using thresholding, topological, and geometric constraints, then liver segmentation is initialized by placing a liver model obtained from the visible human data set on the image and deforming it. For livers that have a shape similar to the liver model, this approach gives rise to good segmentation results. However, for livers that have an atypical shape (about 5 out of the 35 volumes tested), interactive adjustment of the initial model is required to produce acceptable results. The application of the deformable model in the visualization of hepatic vasculature and tumors has also been reported by Chou et al [8], without addressing quantitative accuracy of the segmentation. It is understood that Varma et al [9] have developed a semi-automatic procedure to segment liver along with its venous vessel tree from CT images, which a traditional level-set method with an intensity gradient-dependent speed function is used. The algorithm has been tested on seven volumes but no quantitative information on its performance is reported. Recently, a method based on geodesic deformable models combined with a-priori shape information to limit leakage problems into adjacent structures has been proposed [10].
Geometric deformable models have been proposed as an alternative to parametric deformable models such as snakes [11]. In the geometric deformable model, the evolution of an initial contour toward the true object boundary is considered as a front propagation problem. This permits the use of level set and fast marching methods introduced by Sethian [13] to model propagating fronts with curvature-dependent speeds. Following the level set formulation, the boundary of an object is embedded as the zero-level set of a time-dependent function Φ that is governed by the equation ofΦt+F|∇Φ|=0  (1)where F is a speed function specifying the speed at which the contour evolves along its normal direction. As time progresses, the value of the function Φ evolves and the object's boundary is the zero level set of the function. One of the main advantages of this approach over traditional parametric deformable models is the ease with which topological adaptation can be handled. The main challenge of the front propagation methods is to develop an adequate speed function for a specific application. Malladi et al [14] have proposed a general model for image segmentation in which the speed function is defined asF=gI(F0−εκ)  (2)where gI is a term derived from the image itself, which is used to stop the propagation of the contour near desired points such as points with high gradient or pre-specified intensity values, κ represents the curvature of the front and acts as a regularization term, F0 is a constant; and ε is a weighting parameter. A common expression for gI is in the form ofgI=e−α|∇Gσ*I|,  (3)where Gσ*I denotes the convolution of the image with a Gaussian smoothing filter with standard deviation σ. This term is close to 1 over homogeneous areas and tends to zero over regions with large intensity gradients. It is designed to force fronts to propagate at constant speed over homogeneous regions and to slow them down at the boundary of these regions. However, unless hard thresholds are set on the value of gI, i.e. gI is set to zero when the gradient exceeds a certain value), contours never stop completely. Given enough time, contours will eventually “leak” outside the desired regions. This problem has been recognized and a solution has been proposed [15] by introducing a new term called the doublet term ∇(β|∇Gσ*I|)·{right arrow over (N)} into the level set formulation. The doublet term attracts contours toward boundaries. However, when a boundary is crossed, the doublet term also pulls back the contours towards the boundary. Although the doublet term offers a partial solution to the leakage problem, it is not very well adapted to situations with weak edges in noisy and non-uniform images. Indeed, to stop the front of the contour permanently at the structure boundary, the doublet term needs to be strong enough to offset the constant inflation term (i.e., the constant term F0). To stop the contour at weak edges, this requires the selection of a large β value, but a large value of β tends to stop the contour at spurious edges within the structure. Finding the appropriate value for the parameter β is difficult and often image dependent, which affects the robustness of algorithms that use this type of speed functions. Siddiqi et al [16] further address the leaking problem by introducing a speed function that combines length minimizing and area minimizing terms. It can prevent leakage over small gaps but is of limited use for large gaps in the edge map. On the other hand, speed functions that depend directly or indirectly on the gradient of the images are challenged by images in which object boundaries are ill-defined.
Several methods that are less dependent on the gradient of the images have been proposed recently. These methods typically integrate regional and/or shape constraints into the traditional geometric model. Suri et al [17] have introduced an additional propagation force that is based on regional fuzzy clustering. Paragios [18] has proposed a new framework called geodesic active region, which introduces an energy function incorporates both boundary probability and regional probability that are based on a Bayesian model. Chan et al [19] have also proposed a level-set approach in which the functional being minimized does not depend on edges but on the distribution of intensities within the contours. A similar regional geometric model segmenting images via coupled curve evolution equations has recently been proposed by Yezzi et al [20]. The merit of these methods lies in the fact that the curve evolution, which also can be viewed as pixel/voxel classification, is controlled by competition between different classes in the feature domain (intensity, texture etc.). As a result, these algorithms are capable of segmentation of images with very weak edges or even no edges (gradient boundary). The other advantage of regional based methods is that the initial positions are not as sensitive as gradient methods. However, these methods are limited in medical applications such as liver segmentation where livers are connected to tissues with similar intensity and texture, and are surrounded by structures with vastly different features. Another interesting method proposed by Leventon et al [21] and Tsai et al [22] incorporates global shape constraints into the geometric active contour model. These authors create shape models by performing principle component analysis on level set representations of the training set, which allows them to constrain the segmentation process by integrating global prior shape information. However, the effectiveness of these methods on organs with large inter-subject variations, such as liver, remains to be determined.
Therefore, a heretofore unaddressed need exists in the art to address the aforementioned deficiencies and inadequacies.